# Homework Help: Integral bounds

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1. Dec 3, 2014

### dylanhouse

1. The problem statement, all variables and given/known data
I am having trouble setting up the bounds on the following two integrals:

(a) The region E bounded by the paraboloid y=x2+z2 and the plane y=4.
(b) The region bounded by the cylinder x2+y2=1, z=4, and the paraboloid z=1-x2-y2.

2. Relevant equations

3. The attempt at a solution
I thought for (a) to use 4 < y < x2+z2
-sqrt(y-z2) < x < sqrt(y-z2)
-sqrt(y-x2) < z < sqrt(y-x2)
But these don't seem right.
I'm not sure where to begin for (b) except that z may be upper bounded by 4?

2. Dec 3, 2014

### Orodruin

Staff Emeritus
(a) The conditions you put on y will not give you a bounded region. For simplicity, I suggest you start working with $r=\sqrt{x^2 + z^2}$ instead of x and z.

(b) As in (a), polar coordinates will serve you well here.

3. Dec 3, 2014

### pasmith

For these you want to use cylindrical coordinates. For (a), you should take $(x,y,z) = (r \cos \theta, y, r \sin \theta)$. For (b), you should take $(x,y,z) = (r \cos\theta, r \sin \theta, z)$.